Disclaimer : I am new to this site so I apologize in advance for the mistakes I could eventually make while posting this.
Intuitively, my question is about finding the maximal subsets of a finite grid in terms of cardinal such that their points all satisfy some condition (here they must have some distance between one another).
Formally, it comes down to calculating this :
$Argmax(\{Card(A), A \subset [[0,n[[ \times [[0,m[[ \text{ } \land \text{ } \forall x,y \text{ } \in A \text{ } |x-y| \geq d \})$, $n,m \in \mathbb{N}^*$, $d \in \mathbb{R}^*_+$
Also just to be clear, what I call $Argmax$ is the set of maximal arguments satisfying some property and not one element of it.
This is a question I asked myself out of pure curiosity, but it seems to be one of those hard problems in discrete optimization that we lack tools to solve properly and to be completely honest, I have no clue as to how to solve it either. I've tried to find literature about it but so far none of what I've found seem to present techniques that could get me closer to some solution.
I could of course "bruteforce" my way through the problem by instanciating $n, m$ and $d$ but that would be missing the actual point I'm trying to make.
Is this problem even solvable to begin with ? I'm looking forward to your answers.
Thank you in advance